The vertex of this parabola is at (2, -1). when the y-value is 0, the x-value is 5. what is the coefficient of the squared term in the parabola's equation?

The coefficient would be 1/9.

We first write this in vertex form:
y=a(x-h)²+k, where (h, k) is the vertex.  Using our vertex, we have:

y=a(x-2)²-1

We know that the parabola goes through (5, 0), so we can use these coordinates to find the value of a:
0=a(5-2)²-1
0=a(3)²-1
0=9a-1

Add 1 to both sides:
0+1=9a-1+1
1=9a

Divide both sides by 9:
1/9 = 9a/9
1/9 = a